Protein Structure Similarity

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Protein Structure Similarity
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Computation of Best Matches

• Two simultaneous subproblems
• Find maximal correspondence set C
• Find alignment transform T
• Chicken-and-egg issue
• Each subproblem is relatively simple
• If we knew C, we could compute T
• If we knew T, we could get C by proximity
• But the combination is hard !!!

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Computation of Best Matches

• Two simultaneous subproblems
• Find maximal correspondence set C
• Find alignment transform T
• Chicken-and-egg issue
• Each subproblem is relatively simple
• If we knew C, we could compute T
• If we knew T, we could get C by proximity
• But the combination is hard !!!

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Find Alignment Transform

• Two sets of points A a1,,an and B
  b1,,bn
• Correspondence pairs (ai, bi)
• Find T arg minT RMSD(A,T(B)) ?
• O(n) closed-form solution Arun, Huang, and
  Blostein, 87 Horn, 87 Horn, Hilden, and
  Negahdaripour, 88

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O(n) SVD-Based Algorithm

• T combines translation t and rotation R, such
  that T(bi) t R(bi)
• b (Si1,...,nbi)/n mean of the bis
• Place the origin of coordinate system at b
• minT RMSD(A,T(B)) simplifies to (up to some
  constants)
• t and R can be computed separately
• t a mean of the ais

Arun, Huang, and Blostein, 87
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O(n) SVD-Based Algorithm

• A3?n a1-a, ..., an-a B3?n b1-b, ...,
  bn-b
• Compute SVD decomposition of 33 correlation
  matrix BAT BAT UDVT
  where D is a diagonal matrices with decreasing
  non-negative entries (singular values) along the
  diagonal
• If det(U)det(V) 1 then S I,
  else S diag(1,1,-1)
• R USVT

Arun, Huang, and Blostein, 87
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O(n) SVD-Based Algorithm

• A3?n a1-a, ..., an-a B3?n b1-b, ...,
  bn-b
• Compute SVD decomposition of 33 correlation
  matrix BAT BAT UDVT
  where D is a diagonal matrices with decreasing
  non-negative entries (singular values) along the
  diagonal
• If det(U)det(V) 1 then S I,
  else S diag(1,1,-1)
• R USVT

Arun, Huang, and Blostein, 87
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• Arun, Huang, and Blostein, 87
• ? rotation matrix
• Horn, 87 ? quaternion

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? Trial-and-Error Approach to Protein Structure
Comparison
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? Trial-and-Error Approach to Protein Structure
Comparison

• Set CS to a seed correspondence set (small set
  sufficient to generate an alignment transform)
• Compute the alignment transform T for CS and
  apply T to the second protein B
• Update CS to include all pairs of features that
  are close apart
• If CS has changed, then return to Step 2 else
  return (CS,T)

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? Trial-and-Error Approach to Protein Structure
Comparison

• - result nil
• - Iterate N times
• Set CS to a seed correspondence set (small set
  sufficient to generate an alignment transform)
• Compute the alignment transform T for CS and
  apply T to the second protein B
• Update CS to include all pairs of features that
  are close apart
• If CS has changed, then return to Step 2 else
  result ? result ? (CS,T)
• - Return result

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• How to get seed correspondences?

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Seed Generation from Fragment

• From distance matrices
• E.g., DALI Holm and Sander, 1996

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Using Distance Matrices (DALI)

• Distances are invariant to rigid-body
  transformations
• DALI Holm and Sander, 1996 looks for similar
  hexapeptides by searching for similar 7x7 Ca-Ca
  distance matrices

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Seed Generation from Fragment

• From distance matrices
• E.g., DALI Holm and Sander, 1996
• From secondary structure elements (SSEs)
• E.g., LOCK Singh and Brutlag, 1996
• From voting scheme (using geometric hashing)
• E.g., 3dSEARCH Singh and Brutlag, 2000

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LOCK

• A.P. Singh and D.L. Brutlag. Hierarchical
  Protein Structure Superposition Using Both
  Secondary and Atomic Representations. Proc. ISMB,
  pp. 284-293, 1997.
• LOCK2J. Shapiro and D.L. Brutlag. FoldMiner
  Structural Motif Discovery Using an Improved
  Superposition Algorithm. Protein Science,
  13278-294, 2004.
• http//motif.stanford.edu/lock2/

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LOCK

• Two levels of features SSEs and Ca atoms
• Stage 1 (SSE alignment) Initial alignment is
  computed using SSEs represented as vectors
• Stage 2 (atom alignment) Alignment is refined
  using Ca atoms represented as points

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Rationale for LOCK

• Using types of features is an effective way to
  reduce combinatorial explosion and computation
• SSEs, which are responsible for most of the
  stability and functionality of the proteins, are
  more meaningful and better conserved than types
  of atoms and amino-acids
• If 2 structures are similar, some of their SSEs
  should form similar substructures
• Drawback It narrows down the set of possible
  applications, e.g., cant find small motifs at
  atomic level

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Vector-Based Representation
b-strands
loops
a-helices
One vector per SSE (helix, strand, loop)
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Vector-Based Representation

• DSSP Kabsch and Sander, 1983 classifies
  residues into helices/strands
• For a-helix starting at residue iXorigin
  (0.74Xi Xi1 Xi2 0.74Xi3)/3.48where Xi
  is the position of the Ca atom of residue i
• (angle between two consecutive residues is 100dg
  ? factor 0.74)
• Similar computation for Xend and for b-strand

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Scoring Similarity
Maximal score

• Position-independent differences
• angle(i,k)-angle(p,r)
• angle(i,j)-angle(p,q)
• angle(j,k)-angle(q,r)
• distance(i,k)-distance(p,r)
• length(k)-length(r)
• Position-dependent differences
• angle(k,r)
• distance(k,r)
• Scores are additive

Score S S(di)
Value of di forwhich score is 0
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Stage 1 SSE Alignment

• For every pair of SSE vectors of protein A, find
  all pairs of vectors in B that align well using
  orientation-independent scores ? seed
  correspondence sets
• For each correspondence set
• Find alignment transform and apply it to B
• Find correspondence set with maximal score
• (record transform T and correspondence set CS
  that yields maximal score)

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Stage 1 SSE Alignment

• A (i, j, k, l, m)
• B (p, q, r, s, t)
• Seed correspondence (i,p),(j,q)

• Simultaneous gaps in both structures are not
  allowed (not in SCOP2)
• Terminate a path when score of new
  correspondence is negative
• Re-compute new transform with each new
  correspondence (?)

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Stage 2 Atom (Core) Alignment

• Construct correspondence pairs of atoms
• Atom i of A corresponds to atom j of T(B) iff i
  is the closest atom in A to j and j is the
  closest atom in T(B) to i
• The distance between i and T(j) is lt e (3Å)
• Prune correspondence set to largest subset of
  correspondence pairs that follow backbone
  alignment constraint
• Re-compute T to be the transform that minimizes
  the RMSD of the atoms in the correspondence set
• Iterate 1-2-3 until RSMD converges

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Experimental Results

• 685 protein structures from PDB such that each
  pair has less than 25 sequence identity
• 3 families of folds (based on SCOP
  classification) - myoglobins (11 structures)
  20 amino acid identity- TIM barrels (50
  structures)- immunoglobulins (38 structures)
• Goal Given one query protein in each family,
  find the other members of the family (3685
  2055 alignments)
• Method For each query, sort the 685 structures
  by score (computed by LOCK). Select the top k
  proteins. Count members of family (true
  positives) and non-members (false positives)

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Myoglobins (11)
TIM-barrels (50)
Immunoglobulins (38)
True positives False positives
11 0
True positives False positives
40 0
45 1
50 5
True positives False positives
20 0
25 1
30 2
35 11
38 383
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Alignment of 11 Myoglobins
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Alignment of 50 TIM barrels
a-helices in red b-strands in yellow
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Alignments of 31 Immunoglobulins
Only b-strands are shown
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ROC Curves
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Running Time

• 1ms per seed correspondence
• 1h to search 10,000 protein structures
• 100s of days to compare all pairs of proteins
  in PDB
• ? Geometric hashing to speedup stage 1

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Seed Generation from Fragment

• From distance matrices
• E.g., DALI Holm and Sander, 1996
• From secondary structure elements (SSEs)
• E.g., LOCK Singh and Brutlag, 1996
• From voting scheme (using geometric hashing)
• E.g., 3dSEARCH Singh and Brutlag, 2000

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Voting Scheme with Hash Table

• Many-to-many comparison requires a better
  organization of computation to avoid repeating
  the same computation again and again
• Pre-computation Index proteins in hash table
• Query phase Voting scheme using hash table
• Several variants on this theme
  3d-Lookup Holm and Sander, 1995
  3dSEARCH Singh 2002

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Voting Scheme with Hash Table

• Many-to-many comparison requires a better
  organization of computation to avoid repeting the
  same computation again and again
• Pre-computation Index proteins in hash table
• Query phase Voting scheme using hash table
• Several variants on this theme
  3d-Lookup Holm and Sander, 1995
  3dSEARCH Singh 2002

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Indexing Target Structures in Hash Table
(3dSEARCH Singh 2002)

• Hash table 3-D regular grid of cubic bins (2Å)
• For each target structure
• For each pair of vectors (i,j)
• Compute a coordinate system
• Place an entry for each other vectork into the
  bin containing the coordinates of the midpoint of
  the vector (or average of coordinates of origin,
  middle, and end points). Store ID of coordinate
  system ks orientation and type (a or b) in the
  entry.

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v
u
Grid is same for all coordinate systems
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v
v
u
u
Grid is same for all coordinate systems
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Indexing Target Structures in Hash Table
(3dSEARCH Singh 2002)

• Hash table 3-D regular grid of cubic bins (2Å)
• For each target structure
• For each pair of vectors (i,j)
• Compute a coordinate system
• Place an entry for each other vectork into the
  bin containing the coordinates of the midpoint of
  the vector (or average of coordinates of origin,
  middle, and end points). Store ID of coordinate
  system ks orientation and type (a or b) in
  the entry.
• Grid is sparsely occupied ? hash table
• A structure with n SSEs contributes n(n-1)(n-2)
  entries. Each vector is represented (n-1)(n-2)
  times
• 10,000 structures with 10 SSEs each yield 7M
  entries

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Voting Using Hash Table

• Given a query structure
• For each pair of vectors (i,j)
• Compute a coordinate system
• For each other vector k
• Retrieve the bin accessed by this vector and the
  neighboring bins
• For every entry (vector) in those bins that has
  the same orientation and type as k, add a vote
  for the coordinate system stored in the entry
• Sort target structures based on max number of
  votes received by any of its coordinate systems
• ? Small number of target structures. Use LOCK for
  better alignment
• Hours of pure LOCK are reduced to seconds

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Advantages of Voting System

• Very efficient in practice for many-to-many
  comparisons
• Can establish correspondence between partial,
  disconnected substructures
• Parallel implementation is straightforward
• Independent of the order in which vectors are
  considered
• Drawback (?) May establish correspondences that
  do not satisfy the backbone sequence constraint

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Problem 4 Find Pharmacophore in Ligands

• Given
• Collection of N ( 5 to 10) small flexible
  ligands with similar activity (binding at same
  sites)

Benzamidine binding to beta-Trypsin (3ptb)
Inhibitor binding to HIV protease
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Problem 4 Find Pharmacophore in Ligands

• Given
• Collection of N ( 5 to 10) small flexible
  ligands with similar activity (binding at same
  sites)
• A set of low-energy conformations (dozens to few
  hundreds) for each ligand

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Problem 4 Find Pharmacophore in Ligands

• Given
• Collection of N ( 5 to 10) small flexible
  ligands with similar activity (binding at same
  sites)
• A set of low-energy conformations (dozens to few
  hundreds) for each ligand
• Find a substructure (pharmacophore) that has a
  match in at least one conformation of each ligand

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Pharmacophore and Rational Drug Design

• Pharmacophore identification is a form of
  reverse engineering to get a model of a binding
  site
• A pharmacophore can be used to modify ligands
  into more potent drugs and/or to screen large
  databases of ligands for leads

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Three Simultaneous Problems

• Conformations?
• Correspondence?
• Transform?
• But ligands are small molecules

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Software

• DISCO Martin et al., 1993
• DISCOtech and GASP Tripos, Inc.
• CATALYST and HIPHOP Accelrys et al. Green et
  al., 1994 Barnum et al., 1996
• RAPID P.W. Finn, L.E. Kavraki, J.C. Latombe, R.
  Motwani, C. Shelton, S. Venkatasubramanian, and
  A. Yao. RAPID Randomized Pharmacophore
  Identification for Drug Design. Computational
  Geometry Theory and Applications, 10, pp.
  263-272, 1998

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Pairwise Comparison

• Multi-Probe(M1,,MN)
• Extract invariants from M1 and M2 by calling
  Pair-Probe(P1,P2) on every pair of conformations
  of the two ligands
• Test each candidate invariant S obtained at Step
  1 against every ligand Mi, i 3,,N by calling
  Pair-Probe(S,P) on S and each conformation P of Mi

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Pair-Probe

• n smallest number of atoms/features in a
  liganda given constant (0 lt a 1) P1 and P2
  Conformations of two distinct ligands (or
  candidate invariant)
• Pair-Probe(P1,P2)
• Perform s times
• Pick a triplet of atoms at random from P1
• Determine three atoms in P2 congruent to this
  triplet compute the alignment transform T
• Iterate Apply T to P2 determine the atoms in P1
  matching those in P2 update T
• If the number of matching atoms exceed an, then
  return this atom set as a candidate invariant S

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Magnitude of s

• Prpicking 3 atoms in invariant ? a3
• Prfailing to find invariant ? (1 - a3)s
• We want (1-a3)s ? g (g is acceptable
  probability of failure)
• s ? ln(g)/ln(1-a3)
• Since x lt -ln(1-x) for 0 lt x lt 1, we get s ?
  ln(1/g)/a3
• For g 10-2 and a 0.3, we get s ? 180

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Some Results

• 63 to 69 atoms with 10 to 15 torsional degrees
  of freedom
• Feature every non-H atom ? 30 features of 6
  types(atom types)
• Invariant in active conformations 7-atom
  pharmacophore 7-atom scaffolding

conf t(s) 4 5 6 7 8 9
10 11 12 13 14
11 800 44 20 10 5 2 1 0 0 1 0 0
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Fuel for Thoughts
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Idea Many-to-many correspondence may be more
robust
Example Hausdorf distance
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Hausdorf Distance

• Two sets of points A a1,...,an and B
  b1,...,bm in ?k
• dH(A,B) maxa?A minb?B a-b
• DH(A,B) max dH(A,B), dH(B,A)
• Variation for shape similarity?H(A,B) minT
  DH(A,T(B))
• But efficient algorithms only exist for planar
  sets of points

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Other Idea Minimize cost of transforming A into
B

• Old idea
• Graphics Morphing distance
• Computer vision Earth Movers distanceRubner,
  Tomasi, and Guibas, 1998
• Protein similarity
• Isotopic distance Erdmann, 2004

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Structure Alignment Isotopies

• Two curves are isotopic if one can be deformed
  into the other without self-collision
• Example Polygonal curve with n vertices
• One may think of structure alignment as an
  isotopy deforming one structure into the other
• Two structures are similar if the isotopy is
  small

M.A. Erdmann. Protein Similarity from Knot
Theory GeometricConvolution and Line Weavings,
CMU Tech. Rep. CMU-CS-04-138.
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Small Isotopy

• Model a structure as a set of polygonal lines
  (e.g., vertices are Ca atoms)
• Two structures A and B are (T,d)-isotopic if
  there exists an isotopy deforming A into T(B) in
  such a way that no vertices of A moves further
  away than some d from its initial or final
  location

Erdmann 2004
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Similarity Measure

• dT(A,B) inf d A is (T,d)-isotopic to B
• d(A,B) infT dT(A,B)
• d is computable Erdmann,2004
• But as complex as path planning, hence
  exponential in the number of degrees of freedom
• Possibility of approximating d using
  probabilistic roadmaps?

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Topology of Line Weavings
1xis 1nar
a helix axes
M.A. Erdmann. Protein Similarity from Knot
Theory GeometricConvolution and Line Weavings,
CMU Tech. Rep. CMU-CS-04-138.
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? 2 topologically equivalent line weavings
3 equivalent classes for 4 lines
Erdmann 2004
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Another (incorrect) alignment of 1xis and 1nar
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? 2 non-equivalent line weavings
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Why topology is interesting?

• Two conformations may be geometrically close
  (small RMSD) may require a long continuous
  deformation to map one into the other (without
  steric clashes)

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Conclusion

• Automatic computation of structure similarity is
  essential due to the rapid growth of the PDB and
  other molecule (e.g., ligand) libraries
• As the growth of new protein structures outpaces
  that of new folds, detecting structural
  similarity will have to be much more fine-grained
  than it is today
• Biological discoveries will likely lie in local,
  possibly rare structure similarities, rather than
  in global fold-level classification
• Need for better understanding of applications
  and radically new approaches
• Still a lot of work ...